Proof of Kyle Petersen’s Amazing Conjecture Relating the q- and Fibonacci Analogs of n! Doron ZEILBERGER

Let the darga of a polynomial be the sum of its low-degree and (high) degree, for example, darga(2 + q + q^{2} − q^{3}) = 0 + 3 = 3 , darga(q^{2} + q^{3}) = 2 + 3 = 5 , darga(q^{3}) = 3 + 3 = 6 . It is immediate that every symmetric polynomial of darga d can be written, uniquely, as a linear combination of

(1 + q)^{d} , q(1 + q)^{d−2} , q^{2}(1 + q)^{d−4} , …

in symbols

P (q) = ∑_{i=0}^{⌊d/2⌋} γ_{i} q^{i}(1 + q)^{d−2i} ,

and we will call the Kyle Transform, P̂(z), the polynomial, in z,

P̂(z) := ∑_{i=0}^{⌊d/2⌋} γ_{i} z^{i} .

Kyle Petersen([P]) made the following elegant conjecture, that I will shortly prove. Fact: Let A_{n}(z) be the Kyle transform of

[n]! := ∏_{i=1}^{n} \frac{1 − q^{i}}{1 − q} ,

then

A_{n}(−1) = ∏_{i=1}^{n} F_{i} ,

where the F_{i} are the famous Fibonacci numbers defined by F_{0} = 0, F_{1} = 1, F_{i} = F_{i−1} + F_{i−2} for i ≥ 2. Proof. For a general symmetric P (q) we have

P (q) = ∑_{i=0}^{⌊d/2⌋} γ_{i} q^{i}(1 + q)^{d−2i} = (1 + q)^{d} ∑_{i=0}^{⌊d/2⌋} γ_{i} (\frac{q}{(1 + q)^{2}} )^{i} .

Hence

\frac{P (q)}{(1 + q)^{d}} = P̂(\frac{q}{(1 + q)^{2}} ) .

Letting z = \frac{q}{(1+q)^{2}} , we get the quadratic equation zq^{2} + (2z − 1)q + z = 0 , whose (unique formal-power series) solution is

q = \frac{1 − 2z − \sqrt{1 − 4z}}{2z} = C(z) − 1 = zC(z)^{2} ,

where

C(z) = \frac{1 − \sqrt{1 − 4z}}{2z} ,

is the unique formal power series solution of C(z) = 1 + zC(z)^{2} of Catalan fame. Hence

P̂(z) = \frac{P (zC(z)^{2})}{(1 + zC(z)^{2})^{d}} = \frac{P (zC(z)^{2})}{C(z)^{d}} .

Plugging-in z = −1 we get

P̂(−1) = \frac{P (−C(−1)^{2})}{C(−1)^{d}} .

Now

C(−1) = \frac{1 − \sqrt{5}}{−2} = \frac{\sqrt{5} − 1}{2} = ϕ^{−1} ,

where ϕ = (\sqrt{5} + 1)/2 is the famous golden ratio. Hence

P̂(−1) = \frac{P (−ϕ^{−2})}{ϕ^{−d}} = ϕ^{d} P (−ϕ^{−2}) .

Applying this to P (q) = [n]!, we get

A_{n}(−1) = ϕ^{n(n−1)/2} ∏_{i=1}^{n} \frac{1 − (−ϕ^{−2})^{i}}{1 + ϕ^{−2}} = ∏_{i=1}^{n} \frac{ϕ^{i} − (−ϕ^{−1})^{i}}{ϕ + ϕ^{−1}} = ∏_{i=1}^{n} F_{i} .

Reference

[P] Kyle Petersen, Private communication (via Richard P. Stanley) during the banquet in honor of Ira Gessel, Brandeis University, May 8, 2015.

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: zeilberg at math dot rutgers dot edu .

May 11, 2015; Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger.